A078709 a(n) = floor(n/d(n)), where d(n) is the number of divisors of n (A000005).

1, 1, 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 3, 3, 8, 3, 9, 3, 5, 5, 11, 3, 8, 6, 6, 4, 14, 3, 15, 5, 8, 8, 8, 4, 18, 9, 9, 5, 20, 5, 21, 7, 7, 11, 23, 4, 16, 8, 12, 8, 26, 6, 13, 7, 14, 14, 29, 5, 30, 15, 10, 9, 16, 8, 33, 11, 17, 8, 35, 6, 36, 18, 12, 12, 19, 9, 39, 8, 16, 20, 41, 7, 21, 21, 21, 11

Offset: 1

Author

Joseph L. Pe, Dec 19 2002

Keywords

• nonn

Comments

Also, integer part of the mean subinterval length in the partition of [0,n] by the divisors of n.

If the first occurrence of m in the sequence is greater than all preceding terms, the corresponding n is noncomposite. - Donald Sampson (Marsquo(AT)hotmail.com), Dec 10 2003

Examples

			The divisors of 9 partition the closed interval [0,9] into subintervals [0,1), [1,3), [3,9], with lengths 1, 2, 6, respectively. The mean of these lengths has integer part = 3. Hence a(9) = 3.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A078710.

Programs

Mathematica

<< Statistics`DescriptiveStatistics` f[n_] := Module[{d, l, a, i}, d = Divisors[n]; l = Length[d]; a = {1}; For[i = 1, i <= l - 1, i++, a = Append[a, d[[i + 1]] - d[[i]]]]; a]; Table[Floor[Mean[f[i]]], {i, 1, 100}]
Table[Floor[n/DivisorSigma[0,n]],{n,90}] (* Harvey P. Dale, Jun 10 2016 *)

Python

from sympy import divisor_count
def A078709(n): return n//divisor_count(n) # Chai Wah Wu, Jun 03 2022

Extensions

Replaced definition with a simpler definition suggested by Reinhard Zumkeller, Feb 26 2003. The original definition is now a comment. - N. J. A. Sloane, Jun 19 2022